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Analyzing problems involving definite integrals

AP.CALC:
CHA‑4 (EU)
,
CHA‑4.D (LO)
,
CHA‑4.D.1 (EK)
,
CHA‑4.D.2 (EK)
,
CHA‑4.E (LO)
,
CHA‑4.E.1 (EK)
The interpretation of definite integrals as accumulation of quantities can be used to solve various real-world word problems.
Accumulation (or net change) problems are word problems where the rate of change of a quantity is given and we are asked to calculate the value the quantity accumulated over time. These problems are solved using definite integrals. Let's see how it's done.

Accumulation problems are solved using definite integrals

Imagine we are given the following information:
The temperature of a soup is increasing at a rate of r, left parenthesis, t, right parenthesis, equals, 30, e, start superscript, minus, 0, point, 3, t, end superscript degrees Celsius per minute (where t is the time in minutes). At time t, equals, 0, the temperature of the soup is 23 degrees Celsius.
And imagine we are asked to find the amount by which the temperature increased between t, equals, 0 and t, equals, 5 minutes. This is an accumulation (or net change) word problem. We can tell it is so because we are given a function that models the rate of change of a quantity, and are asked about the change in that quantity over an interval of time.
For any quantity whose rate is given by the function r, the definite integral integral, start subscript, a, end subscript, start superscript, b, end superscript, r, left parenthesis, t, right parenthesis, d, t describes the amount by which the quantity changed between t, equals, a and t, equals, b.
So in our case, the amount by which the temperature increased between t, equals, 0 and t, equals, 5 minutes is given by integral, start subscript, 0, end subscript, start superscript, 5, end superscript, r, left parenthesis, t, right parenthesis, d, t.
integral, start subscript, 0, end subscript, start superscript, 5, end superscript, r, left parenthesis, t, right parenthesis, d, t, approximately equals, 77, point, 7 degrees Celsius
Now imagine we were asked a different question: what is the soup's temperature at t, equals, 5 minutes? Notice we aren't dealing with a change anymore, we're dealing with an actual value. But have no fear, because definite integrals can help us with this one too! All we need is to add the initial condition.
Recall we were given that the temperature of the soup at time t, equals, 0 was 23 degrees Celsius. Adding that to the change in temperature between t, equals, 0 and t, equals, 5 gives us the temperature at t, equals, 5:
start underbrace, 23, end underbrace, start subscript, start text, t, e, m, p, space, a, t, space, end text, t, equals, 0, end subscript, plus, start overbrace, integral, start subscript, 0, end subscript, start superscript, 5, end superscript, r, left parenthesis, t, right parenthesis, d, t, end overbrace, start superscript, start text, c, h, a, n, g, e, space, i, n, space, t, e, m, p, space, f, r, o, m, space, end text, t, equals, 0, start text, space, t, o, space, end text, t, equals, 5, end superscript
Since we already calculated integral, start subscript, 0, end subscript, start superscript, 5, end superscript, r, left parenthesis, t, right parenthesis, d, t, we can tell that at t, equals, 5 minutes, the temperature was 23, plus, 77, point, 7, equals, 100, point, 7 degrees Celsius. That's boiling hot!
Problem 1
Consider the following problem:
The population of a town grows at a rate of r, left parenthesis, t, right parenthesis, equals, 300, dot, e, start superscript, 0, point, 3, t, end superscript people per year (where t is time in years). At time t, equals, 2, the town's population is 1200 people. What is the town's population at t, equals, 7?
Which expression can we use to solve the problem?

Common mistake: Misusing initial conditions

Some accumulation problems ask about a net change, and some ask about an actual value. The difference is that when looking for an actual value we must use the initial conditions.
A common mistake would be using initial conditions when asked about a net change, or not using initial conditions when asked about an actual value.

Common mistake: Using differentiation instead of integration

Applied word problems are common throughout both differential and integral calculus. When given a word problem, we must decide whether the solution involves derivatives or integrals. Making a wrong decision will of course result in a wrong answer.
Derivatives are useful when we are given a quantity and asked about its rate, while integrals are useful when we are given a rate and asked about the quantity.
What's given?What's missing?What to use?
Differential calculusQuantityRateDerivative
Integral calculusRateQuantity (or change in quantity)Integral
Problem 2
Consider the following problem:
The depth of the water in a tank is changing at a rate of r, left parenthesis, t, right parenthesis, equals, 0, point, 3, t centimeters per minute (where t is the time in minutes). At time t, equals, 0, the depth of the water is 35 centimeters. What is the change in the water's depth during the fourth minute?
Which expression can we use to solve the problem?